The introduction of computational grids has resulted in several new problems in the area of scheduling that can be addressed using predictions. The first problem is selecting where to run an application on the many resources available in a grid. Our approach to help address this problem is to provide predictions of when an application would start to execute if submitted to specific scheduled computer systems. The second problem is gaining simultaneous access to multiple computer systems so that distributed applications can be executed. We help address this problem by investigating how to support advance reservations in local scheduling systems. Our approaches to both of these problems are based on predictions for the execution time of applications on space- shared parallel computers. As a side effect of this work, we also discuss how predictions of application run times can be used to improve scheduling performance.